388 FOURTH REPORT — 1834. 



velocity and direction of the wave he deduces those of the ray, 

 and therefore the form of the wave- surface, by means of the 

 general relations suggested by his view of mathematical optics. 



In this system, of which the author gave a brief sketch 

 at the late meeting of the Association, the laws of reflexion 

 and refraction, ordinary or extraordinary, are compi'ised in 

 two fundamental expressions, which state that the partial dif- 

 ferential coefficients of the first order of a certain function, — 

 taken with respect to two final coordinates in the plane which 

 touches the reflecting or refracting surface at the point of inci- 

 dence, — are not altered by reflexion or refraction. The function 

 here considered is the characteristic function of the author, 

 whose particular form may be considered as characterizing the 

 optical system, and on whose properties, he finds, all the pro- 

 blems of mathematical optics may be made to depend. On the 

 principles of the wave-theory, this function is equal to the undu- 

 latory time of propagation of light, from any one assumed point 

 to another, in the same or in a difi"erent medium; and the ex- 

 pressions just alluded to, signify simply that the components of 

 normal sloivness of the wave parallel to the bounding surface, 

 or the reciprocal of the velocity of wave-propagation resolved in 

 the direction of that surface, are not changed by reflexion or 

 refraction. The normal slowness of wave-propagation is, then, 

 of fundamental importance in this theory ; and if it be repre- 

 sented in magnitude by a line taken in its direction, there is 

 obtained for its expression a cvirved surface which, on the prin- 

 ciples of Fresnel, is found to be a surface of two sheets, connected 

 with the wave-surface by a remarkable relation of reciprocity. 

 When this relation is combined with the laws of reflexion and 

 refraction just alluded to, they lead to a very elegant construc- 

 tion for the reflected or refracted ray which is, in most cases, 

 more convenient than that of Huygens. Thus, when a ray pro- 

 ceeds from air into any crystal, we have only to construct the 

 surfaces ofivave-sloivness belonging to the two media, and having 

 their common centre at the point of incidence. Let the incident 

 ray be then produced to meet the sphere, which represents the 

 normal slowness of the wave in air ; and from the point of in- 

 tersection let a perpendicular be drawn to the reflecting or refract- 

 ing surface. This will cut the surface of slowness of the reflected 

 or refracted waves in general in two points. The lines connect- 

 ing these points with the centre, will represent the direction and 

 normal slowness of the waves ; while the perpendiculars from 

 the centre on the tangent planes at the same points, will repre- 

 sent the direction and slowness of the ?•««/« themselves. 



This important curved surface presented itself also to M. Cau- 



